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  "body": "\n## Definition\n\nLet $F$ be a field and let $A$ be an $m \\times n$ matrix over $F$ with rows $r_1, \\dots, r_m \\in F^n$.\nThe *row space* of $A$ is\n$$\\operatorname{Row}(A) = \\operatorname{span}(r_1, \\dots, r_m) \\subseteq F^n,$$\nthe [[linear span|0DTY8B9HMS]] of the rows.\n\n## Remarks\n\nThe row space of $A$ is the [[column space|03AJKXH1AV]] of $A^T$.\n[[Elementary row operations|04XWH28452]] do not change the row space, so [[row equivalent|006BHFWMAP]] matrices have the same row space.\nThe nonzero rows of a row echelon form of $A$ form a basis of $\\operatorname{Row}(A)$.\nIts dimension equals the dimension of the column space, see [[row rank equals column rank|0TZAG60ZYT]].\n\n## Explanations\n\n### L1 human\n\nThe row space collects all the combinations of the rows of a matrix.\nRow operations only mix the rows, so they keep the row space.\nAfter row reduction, the nonzero rows give a basis of it.\n",
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